Two-term relative cluster tilting subcategories, $τ-$tilting modules and silting subcategories

Abstract: Let $\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\mathcal{R}$ a rigid subcategory of $\mathcal{C}$. We introduce the notions of two-term $\mathcal{R}[1]$-rigid subcategories, two-term (weak) $\mathcal{R}[1]$-cluster tilting subcategories and two-term maximal $\mathcal{R}[1]$-rigid subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\mathcal{R}[1]$-rigid subcategories of $\mathcal{C}$ and the set of $\tau$-rigid subcategories of $\mod\mathcal{R}$, which induces a one-to-one correspondence between the set of two-term weak $\mathcal{R}[1]$-cluster tilting subcategories of $\mathcal{C}$ and the set of support $\tau$-tilting subcategories of $\mod\mathcal{R}$. This generalizes the main results in \cite{YZZ} where $\mathcal{R}$ is a cluster tilting subcategory. When $\mathcal{R}$ is a silting subcategory, we prove that the two-term weak $\mathcal{R}[1]$-cluster tilting subcategories are precisely two-term silting subcategories in \cite{IJY}. Thus the bijection above induces the bijection given by Iyama-J{\o}rgensen-Yang in \cite{IJY}